AMS 507 Sample Final Exam
1. N dice are tossed at the same time. The smallest number is recorded as X.
a. Find the distribution of X.
Let Yi be the number appearing on the ith die, i=1, 2, ., N.
Since X is the smallest number, it is appropriate to find P(
Time Series Lecture 13: Frequency Domain
CMU MSP 36618 April 3, 2014
I.
Introduction to Spectral Analysis
a. Spectral Analysis (SAS PROC SPECTRA, R spectrum() or spec.pgram() uses regression
on a set of Fourier basis functions to represent any time series
LectureNotes#1
AQuickReviewofProbability&Statistics
(1)
ReviewofProbability,theMontyHallProblem
(http:/en.wikipedia.org/wiki/Monty_Hall_problem)
The Monty Hall problem is a probability puzzle loosely based on the American
televisiongameshowLet'sMakeaDeala
AMS586 Si Wen
Report for Fitting a Time Series Model
1. Introduction
In this report, I choose the stock price of Starbucks Corporation
(SBUX) as my data to analyze. The time period for the data is from
Jan 2009 to Dec 2013. The total number of data points
AMS316 Homework 4 Solution
3.6 (a) Consider the equation y 2 1 y 2 = 0, its roots are given by
y1,2 =
1
2 + 42
1
.
2
The stationarity requires that |y1,2 | < 1. When 2 +42 0, we have
1
2
1 < y1,2 < 1; when 1 + 42 < 0, we have |y1,2 |2 < 1. Simplifying
th
AMS316 HW4
3.6 Find the values of 1 , 2 , such that the second-order AR process dened by
Xt = 1 Xt1 + 2 Xt2 + Zt
is stationary. If 1 = 1/3, 2 = 2/9, show that the ac.f of Xt is given
by
16 2 |k|
1 |k|
5
(k) =
+
, k = 0, 1, 2, .
21 3
21
3
3.7 Explain what
Lecture Notes #1
A Quick Review of Probability & Statistics
(1)
Normal Distribution
Q. Who invented the normal distribution?
* Left: Abraham de Moivre (26 May 1667 in Vitry-le-Fran Champagne,
ois,
France 27 November 1754 in London, England) *From the Wiki
AMS 316 Time Series Analysis
Fall, 2014
Lecture Notes: Sum of Independent ARMA Processes
Let
and
be two ARMA time series that are independent at all
leads and lags, then we have the following results.
1. MA(1) + White noise = MA(1)
For
According
, we have
Simple Linear
Regression
1
1. Introduction
Example:
David Beckham: 1.83m
Victoria Beckham: 1.68m
Brad Pitt:
1.83m
Angelina Jolie: 1.70m
George Bush :1.81m
Laura Bush: ?
To predict height of the wife in a couple, based on the husbands height
Response (out
AMS 507 Test 2
Name: _
SBU ID _
1. Let X be a Poisson random variable with parameter . What value of maximizes
Pcfw_X=k, k0?
Solution:
g ( ) = P ( X = k ) = e
g ' ( ) =
e
(k
k 1
k
k!
k
k!
)=
k 1
e
(k ) = 0
k!
Critical point: = k .
When <k, g'>0. When >k
AMS 507 Test 1
Name: _
SBU ID _
1. Assume that each child who is born is equally likely to be a boy or a girl.
If a family has two children, what is the probability that both are girls given that
(a) the eldest is a girl, (b) at least one is a girl?
Solut
AMS 507
Introduction to Probability
Fall Semester, 2013
Class: Tuesdays, Thursdays, 11:30-12:50, Engineering 145
Instructor: Eugene Feinberg
Office Hours: Tuesdays & Thursdays: 10:00-11:00 and by appointment. My office is in
the Mathematics Tower, Room 1-
AMS 507 Sample Test 1
1. Suppose that a jar contains 2N balls, two of which are marked 1, two marked 2, two
marked 3, ., and two marked N. Draw m cards at random without replacement, where
m 2N 2. For N = 10 and m = 4, nd the probability that the pair of
AMS 507 Sample Test 2
1. Let X be a uniform distribution on (-1,1). Find the density of Y=|X|.
Solution:
f(x)=1/2, -1<x<1.
F y P X y P y X y y
f y 1,0 y 1.
2. Let X be an exponential random variable. Show that P(X>s+t | x>s)=P(X>t) for s>0,
t>0. This is k
Multiple Linear Regression
and
the General Linear Model
1
Outline
1. Introduction to Multiple Linear Regression
2. Statistical Inference
3. Topics in Regression Modeling
4. Example
5. Variable Selection Methods
6. Regression Diagnostic and Strategy for
Bu
Time series Analysis
ARCH/GARCH models
Ruibo Yang , Ensi Qi , Shangyang Tang , Si Wen
Risk Management
Risk: the quantifiable likelihood of loss or less-thanexpected returns
In recent decades the field of financial risk management has
undergone explosive
AMS586 Si Wen
Report for Fitting a Time Series Model
1. Introduction
In this report, I choose the stock price of Starbucks Corporation
(SBUX) as my data to analyze. The time period for the data is from
Jan 2009 to Dec 2013. The total number of data points
AMS316 HW3
In the following equations, cfw_Zt is a discrete-time, purely random pro2
cess such that E(Zt ) = 0, Var(Zt ) = Z , and successive values of Zt are
independent so that Cov(Zt , Zt+k ) = 0, k = 0.
3.1 Show that the ac.f of the second-order MA p
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Mobile
Why Linux?
Other
3%
Windows
3%
Android
46%
iOS
48%
Server/Supercomputer/Cluster
Windows
4%
Linux
79%
Unix
17%
Desktop
Linux
1%
Windows
92%
Mac
7%
Linux family tree
Try uname -a to show your distribution
Excise 1: Enter these commands at the Linux
AMS595 Final project
Numerical Integration
Try to integrate a function numerically:
b
f (x), x [a, b]
F (a, b) =
a
Here we dene f (x) =
1
,x [0, 2]
1 + x2
Requirements:
1. Try to use Midpoint Rule, Trapezoidal Rule, and Simpsons Rule to
implement it numer
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